Evaluate the integral. ∫(e5x+103x+sin⁡4x)dx

Question

Evaluate the integral.  ∫(e5x+103x+sin⁡4x)dx

Fommeirj · Accepted Answer

Step 1
To Determine: evaluate the integral
Given: we have given an integral
Explanation: we have an integral

\int (e^{5 x} + 10^{3 x} + \sin 4 x) d x

We know that

\int a^{x} d x = \frac{a^{x}}{\log a} and a l s o \int e^{5 x} d x = \frac{1}{5} e^{5 x}

So we have
Step 2

\frac{1}{5} e^{5 x} + \frac{10^{3 x}}{3 \log 10} - \frac{1}{4} \cos 4 x

\frac{1}{5} e^{5 x} + \frac{1000^{x}}{3 \log 10} - \frac{1}{4} \cos 4 x

George Burge · Accepted Answer

Step 1: Expand.  ∫(e5x+103x+sin⁡4x)dx  Step 2: Use Sum Rule: ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx.  ∫e5x+∫103xdx+∫sin⁡4xdx  Step 3: Use Integration by Substitution on ∫e5xdx.  Let u=5x, du=5dx, then dx=15du  Step 4: Using u and du above, rewrite ∫e5xdx.  ∫eu5du  Step 5: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  15∫eudu  Step 6: The integral of ex is ex.  eu5  Step 7: Substitute u=5x back into the original integral.  e5x5  Step 8: Rewrite the integral with the completed substitution.  e5x5+∫103xdx+∫sin⁡4xdx  Step 9: Use Integration by Substitution on ∫103xdx.  Let u=3x, du=3dx, then dx=13du  Step 10: Using u and du above, rewrite ∫103xdx.  ∫10u3du  Step 11: Use Constant Factor Rule: ∫cf(x)dx=c∫f(x)dx.  13∫10udu  Step 12: Use this property: ∫axdx=axln⁡a.  10u3ln⁡10  Step 13: Substitute u=3x back into the original integral.  103x3ln⁡10  Step 14: Rewrite the integral with the completed substitution.

Evaluate the integral. \int (e^{5x}+10^{3x}+\sin 4x)dx

Answered question

Answer & Explanation

New Questions in Integral Calculus